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14   LOCALLY CONVEX SPACES       63

For it is clear thatX^*) = MX*); also we have

pa(x + y) g pa(jc) + pa(y)  p(x) + p(y)
for all a E I, hence X* + y) ^ p(jc) + p(y).

(12.14.2)    Let p be a seminorm on a topological vector space E. Then the
following conditions are equivalent:

(a)    p is continuous;

(b)    there exists a neighborhood V ofOon which p is bounded.

Clearly (a) implies (b). Conversely, if p(x) g a for all xe V, then for all
;c0 e E and all x e x0 + fiV we have \p(x) - X*o)l = X*  *o) = a> an(* there-
fore p is continuous on E.

(12.14.3)    Let (px)a e i 6e any family ofseminorms on a vector space E. Then the
topology defined by the pseudo-distances da(x,y)  pjix  y) (12.4) is com-
patible with the vector space structure ofE.

In view of the definitions, the proofs are the same as in (4.1.1) and (5.1.5).

The topology defined by the pseudo-distances d^ is said to be defined by the
family ofseminorms (/?)6i. Two families of seminorms on E are said to be
equivalent if they define the same topology (cf. (12.14.12)). A topological
vector space whose topology can be defined by a family ofseminorms is said
to be locally convex. If (/>a)aei *s a farcnty ofseminorms defining the topology
of a locally convex space E, we obtain a fundamental system of neighborhoods
of 0 in E by considering the sets

(12.14.3.1)         W((oO; r) - {x e E \pai(x) <r   for    lin}

where (a i)^^,, runs through the set of finite families of indices belonging to I,
and r is any real number >0.

(12.14.4) Let E be a vector space and let (/?a)a e j be a family ofseminorms on E.
For the topology defined by the family (pa) to be Hausdorff, it is necessary and
sufficient that for each x ^ 0 in E there should exist an index a e I such that

P*(X) * 0.

This follows from (12.4.4).

Hence a seminorm p on a vector space E is a norm if and only if the
topology defined by this single seminorm is Hausdorff.is there-